Search

This paper introduces the 3D Peskin problem: a two-dimensional elastic membrane immersed in a three-dimensional steady Stokes flow. We obtain the equations that model this free boundary problem and show that they admit a boundary integral reduction, providing an evolution equation for the elastic interface. We consider general nonlinear elastic laws, i.e. the fully nonlinear Peskin problem, and prove that the problem is well-posed in low-regularity Hölder spaces. Moreover, we prove that the elastic membrane becomes smooth instantly in time. [ABSTRACT FROM AUTHOR]

We study the problem where a one‐dimensional elastic string is immersed in a two‐dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non‐linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time wellposedness for arbitrary initial data in the scaling critical Besov space Ḃ2,13/2(T;R2)$\dot{B}^{3/2}_{2,1}(\mathbb {T}; \mathbb {R}^2)$. We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancelation structure. [ABSTRACT FROM AUTHOR]

Although the nuclear fusion process has received a great deal of attention in recent years, the amount of mathematical analysis that supports the stability of the system seems to be relatively insufficient. This paper deals with the mathematical analysis of the magnetic confinement of the plasma via kinetic equations. We prove the global wellposedness of the Vlasov-Maxwell system in a two-dimensional annulus when a huge (but finite-in-time) external magnetic potential is imposed near the boundary. We assume that the solution is axisymmetric. The authors hope that this work is a step towards a more generalized work on the three-dimensional Tokamak structure. The highlight of this work is the physical assumptions on the external magnetic potential well which remains finite within a finite time interval and from that, we prove that the plasma never touches the boundary. In addition, we provide a sufficient condition on the magnitude of the external magnetic potential to guarantee that the plasma is confined in an annulus of the desired thickness which is slightly larger than the initial support. Our method uses the cylindrical coordinate forms of the Vlasov-Maxwell system. [ABSTRACT FROM AUTHOR]

In this paper, we prove the propagation of uniform upper bounds for the spatially homogeneous relativistic Boltzmann equation. These polynomial and exponential L ∞ bounds have been known to be a challenging open problem in relativistic kinetic theory. To accomplish this, we establish two types of estimates for the gain part of the collision operator. First, we prove a potential type estimate and a relativistic hyper-surface integral estimate. We then combine those estimates using the relativistic counterpart of the Carleman representation to derive uniform control of the gain term for the relativistic collision operator. This allows us to prove the desired propagation of the uniform bounds of the solution. We further present two applications of the propagation of the uniform upper bounds: another proof of the Boltzmann H-theorem, and the asymptotic convergence of solutions to the relativistic Maxwellian equilibrium. [ABSTRACT FROM AUTHOR]

This paper proves the existence of small‐amplitude global‐in‐time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well‐known works [45] and [3, 43] on the construction of classical solutions in smooth Sobolev spaces which in particular are regular in the spatial variables, it still remains an open problem to obtain global solutions in an Lx,v∞ framework, similar to that in [49], for the Boltzmann equation with the cutoff assumption in general bounded domains. One main difficulty arises from the interaction between the transport operator and the velocity‐diffusion‐type collision operator in the non‐cutoff Boltzmann and Landau equations; another major difficulty is the potential formation of singularities for solutions to the boundary value problem. In the present work we introduce a new function space with low regularity in the spatial variable to treat the problem in cases when the spatial domain is either a torus or a finite channel with boundary. For the latter case, either the inflow boundary condition or the specular reflection boundary condition is considered. An important property of the function space is that the LT∞Lv2 norm, in velocity and time, of the distribution function is in the Wiener algebra A(Ω) in the spatial variables. Besides the construction of global solutions in these function spaces, we additionally study the large‐time behavior of solutions for both hard and soft potentials, and we further justify the property of propagation of regularity of solutions in the spatial variables. © 2019 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

In this paper we prove the global existence, uniqueness, optimal large time decay rates, and uniform gain of analyticity for the exponential PDE ht = Δe-Δh in the whole space Rdx. We assume the initial data is of medium size in the Wiener algebra A(Rd); we use the initial condition Δh0 ∊ A(Rd) which is scale-invariant with respect to the invariant scaling of the exponential PDE. This exponential PDE was derived in [18] and more recently in [22]. [ABSTRACT FROM AUTHOR]

Adverse drug events (ADEs), a subset of AEs (adverse events), occur among home care recipients, yet the topic of medication safety in this setting is understudied and under reported. The Safety Outcomes With Home Assessment Trial (SO WHAT) was a mixed-methods, pilot study that sought to evaluate the feasibility of prospective evaluation and inform future research in the home care setting to better understand medication use and safety. Over 21 months, 19 patients were recruited. Feasibility of the study was improved by focused recruitment efforts and connection with primary care. All participants had medication issues identified by the pharmacist’s medication review and almost half had a suspected ADE. Patient experience themes identified related to knowledge about medications, communication, and home care human resources. [ABSTRACT FROM AUTHOR]

We prove time decay of solutions to the Muskat equation in 2D and in 3D. In the previous work, the following norms were utilized to prove global existence of solutions to the Muskat problem: In this paper, for the 3D Muskat problem, given initial datafor somel≥3 such thatfor a constantk0≈1∕5, we prove uniform in time bounds of ∥f∥s(t) for −2

This paper considers the three-dimensional Muskat problem in the stable regime. We obtain a conservation law which provides an L² maximum principle for the fluid interface. We also show global in time existence for strong and weak solutions with initial data controlled by explicit constants. Furthermore we refine the available estimates to obtain global existence and uniqueness for strong solutions with larger initial data than we previously had in 2D. Finally we provide global in time results in spaces with critical regularity, giving solutions with bounded slope and time integrable bounded curvature. [ABSTRACT FROM AUTHOR]

In this paper, for both the sharp front surface quasi-geostrophic equation and the Muskat problem, we rule out the "splash singularity" blow-up scenario; in other words, we prove that the contours evolving from either of these systems cannot intersect at a single point while the free boundary remains smooth. Splash singularities have been shown to hold for the free boundary incompressible Euler equation in the form of the water waves contour evolution problem. Our result confirms the numerical simulations in earlier work, in which it was shown that the curvature blows up because the contours collapse at a point. Here, we prove that maintaining control of the curvature will remove the possibility of pointwise interphase collapse. Another conclusion that we provide is a better understanding of earlier work in which squirt singularities are ruled out; in this case, a positive volume of fluid between the contours cannot be ejected in finite time. [ABSTRACT FROM AUTHOR]

The spatially homogeneous Boltzmann equation has been studied extensively in the Newtonian case, but not much is known for the special relativistic case. In this paper, we address several issues for the spatially homogeneous Boltzmann equation for relativistic particles. We first derive the relativistic version of the Povzner inequality. Using this, we study the Cauchy problem and investigate how the polynomial and exponential moments in L¹ are propagated. Several key differences between the relativistic and the Newtonian cases are confronted and discussed. [ABSTRACT FROM AUTHOR]

For the Landau–Poisson system with Coulomb interaction in $${\mathbb{R}^{3}_{x}}$$ R x 3 , we prove the global existence, uniqueness, and large time convergence rates to the Maxwellian equilibrium for solutions which start out sufficiently close. [ABSTRACT FROM AUTHOR]

We present the uniform L¹-stability estimate for the relativistic Boltzmann equation near vacuum. For this, we explicitly construct a relativistic counterpart of the nonlinear functional which is a linear combination of L¹-distance and a collision potential. This functional measures the L¹-distance between two continuous mild solutions. When the initial data is sufficiently small and decays exponentially fast, we show that the functional satisfies the uniform stability estimate leading to the uniform L¹-stability estimate with respect to initial data. [ABSTRACT FROM AUTHOR]

The Muskat problem models the dynamics of the interface between two incompressible immiscible fluids with different constant densities. In this work we prove three results. First we prove an L2.R/maximum principle, in the form of a new "log" conservation law (3) which is satisfied by the equation (1) for the interface. Our second result is a proof of global existence for unique strong solutions if the initial data is smaller than an explicitly computable constant, for instance kf k1 1=5. Previous results of this sort used a small constant 1 which was not explicit [7, 19, 9, 14]. Lastly, we prove a global existence result for Lipschitz continuous solutions with initial data that satisfy kf0kL1 < 1 and k@xf0kL1 < 1. We take advantage of the fact that the bound k@xf0kL1 < 1 is propagated by solutions, which grants strong compactness properties in comparison to the log conservation law. [ABSTRACT FROM AUTHOR]

In this paper we study the large-time behavior of perturbative classical solutions to the hard and soft potential Boltzmann equation without the angular cut-off assumption in the whole space Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. with n ≥ 3. We use the existence theory of global in time nearby Maxwellian solutions from [12, 11]. It has been a longstanding open problem to determine the large time decay rates for the soft potential Boltzmann equation in the whole space, with or without the angular cut-off assumption [26, 1]. For perturbative initial data, we prove that solutions converge to the global Maxwellian with the optimal large-time decay rate of O(t-n/2+n/2r) in the Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed.-norm for any 2 ≤ r ≤ ∞ [ABSTRACT FROM AUTHOR]

For the relativistic Boltzmann equation in Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. , this work proves the global existence, uniqueness, positivity, and optimal time convergence rates to the relativistic Maxwellian for solutions which start out sufficiently close under the general physical soft potential assumption proposed in 1988 [13]. [ABSTRACT FROM AUTHOR]

In this paper we prove the global in time well-posedness of the following non-local diffusion equation with α ∈ (0, 2/3): The initial condition u 0 is positive, radial, and non-increasing with u 0 ∈ L 1 ∩ L 2+δ(ℝ3) for some small δ > 0. There is no size restriction on u 0. This model problem appears of interest due to its structural similarity with Landau's equation from plasma physics, and moreover its radically different behavior from the semi-linear Heat equation: u t = ▵ u + αu 2. [ABSTRACT FROM AUTHOR]

In this paper it is shown that unique solutions to the relativistic Boltzmann equation exist for all time and decay with any polynomial rate towards their steady state relativistic Maxwellian provided that the initial data starts out sufficiently close in $${L^\infty_\ell}$$. If the initial data are continuous then so is the corresponding solution. We work in the case of a spatially periodic box. Conditions on the collision kernel are generic in the sense of Dudyński and Ekiel-Jeżewska (Commun Math Phys 115(4):607-629, 1985); this resolves the open question of global existence for the soft potentials. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for the relativistic Boltzmann equation with near vacuum initial data. Unique global-in-time mild solutions are obtained uniformly in the speed of light parameter c ≤ 1. We furthermore prove that solutions to the relativistic Boltzmann equation converge to solutions of the Newtonian Boltzmann equation in the limit as c → ∞on arbitrary time intervals [0, T], with convergence rate 1/c2-ε for any ε ∈ (0, 2). This may be the first proof of unique global-in-time validity of the Newtonian limit for a kinetic equation. [ABSTRACT FROM AUTHOR]

This is a brief announcement of our recent proof of global existence and rapid decay to equilibrium of classical solutions to the Boltzmann equation without any angular cutoff, that is, for long-range interactions. We consider perturbations of the Maxwellian equilibrium states and include the physical cross-sections arising from an inverse-power intermolecular potential r-(p-1) with p > 2, and more generally. We present here a mathematical framework for unique global in time solutions for all of these potentials. We consider it remarkable that this equation, derived by Boltzmann (1) in 1872 and Maxwell (2) in 1867, grants a basic example where a range of geometric fractional derivatives occur in a physical model of the natural world. Our methods provide a new understanding of the effects due to grazing collisions. [ABSTRACT FROM AUTHOR]

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in 3 with non-trivial swirl. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies, for some 0 ≤ ε ≤ 1, |v (x, t)| ≤ C* r-1+ε |t|-ε/2 for - T0 ≤ t < 0 and 0 < C* < ∞ allowed to be large. We prove that v is regular at time zero. [ABSTRACT FROM AUTHOR]

We consider both soft potentials with angular cutoff and Landau collision kernels in the Boltzmann theory inside a periodic box. We prove that any smooth perturbation near a given Maxwellian approaches zero at the rate of $$e^{-\lambda t^{p}}$$ for some λ > 0 and 0 < p < 1. Our method is based on an unified energy estimate with appropriate exponential velocity weight. Our results extend the classical result Caflisch of [2] to the case of very soft potential and Coulomb interactions, and also improve the recent “almost exponential” decay results by [5, 14]. [ABSTRACT FROM AUTHOR]

The Balescu-Lenard equation from plasma physics is widely considered to include a highly accurate correction to Landau's fundamental collision operator. Yet so far it has seen very little mathematical study. We perform an extensive linearized analysis of this equation, which includes determining the asymptotic behavior of the new components of the linearized operator and establishing time decay rates for the linearized equation. [ABSTRACT FROM AUTHOR]

The Vlasov–Maxwell–Boltzmann system is one of the most fundamental models to describe the dynamics of dilute charged particles, where particles interact via collisions and through their self-consistent electromagnetic field. We prove existence of global in time classical solutions to the Cauchy problem near Maxwellians. [ABSTRACT FROM AUTHOR]

The relativistic Landau-Maxwell system is the most fundamental and complete model for describing the dynamics of a dilute collisional plasma in which particles interact through Coulombic collisions and through their self-consistent electromagnetic field. We construct the first global in time classical solutions. Our solutions are constructed in a periodic box and near the relativistic Maxwellian, the Jüttner solution. [ABSTRACT FROM AUTHOR]

Consider axisymmetric strong solutions of the incompressible Navier–Stokes equations in with nontrivial swirl. Such solutions are not known to be globally defined, but it is shown in ([1], Partial regularity of suitable weak solutions of the Navier–Stokes equations. Communications on Pure and Applied Mathematics, 35 (1982), 771–831) that they could only blow up on the axis of symmetry. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies the pointwise scale invariant bound v(x, t)  C*(r2 − t)−1/2 for −T0  t < 0 and 0 < C* < ∞ allowed to be large, we then prove that v is regular at time zero. [ABSTRACT FROM AUTHOR]